Elliptically contoured distributions can be considered to be the
distributions for which the contours of the density functions are proportional
ellipsoids. We generalize elliptically contoured densities to ``star-shaped
distributions'' with concentric star-shaped contours and show that many results
in the former case continue to hold in the more general case. We develop a
general theory in the framework of abstract group invariance so that the
results can be applied to other cases as well, especially those involving
random matrices